3.734 \(\int \frac{-1+x^2}{\left (1+x^2\right ) \sqrt{x+x^3}} \, dx\)

Optimal. Leaf size=12 \[ -\frac{2 x}{\sqrt{x^3+x}} \]

[Out]

(-2*x)/Sqrt[x + x^3]

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Rubi [A]  time = 0.111015, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]

[Out]

(-2*x)/Sqrt[x + x^3]

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Rubi in Sympy [A]  time = 12.4373, size = 15, normalized size = 1.25 \[ - \frac{2 \sqrt{x^{3} + x}}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-1)/(x**2+1)/(x**3+x)**(1/2),x)

[Out]

-2*sqrt(x**3 + x)/(x**2 + 1)

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Mathematica [A]  time = 0.0118042, size = 12, normalized size = 1. \[ -\frac{2 x}{\sqrt{x^3+x}} \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^2)/((1 + x^2)*Sqrt[x + x^3]),x]

[Out]

(-2*x)/Sqrt[x + x^3]

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Maple [A]  time = 0.006, size = 11, normalized size = 0.9 \[ -2\,{\frac{x}{\sqrt{{x}^{3}+x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-1)/(x^2+1)/(x^3+x)^(1/2),x)

[Out]

-2*x/(x^3+x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} - 1}{\sqrt{x^{3} + x}{\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)),x, algorithm="maxima")

[Out]

integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)), x)

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Fricas [A]  time = 0.270438, size = 22, normalized size = 1.83 \[ -\frac{2 \, \sqrt{x^{3} + x}}{x^{2} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)),x, algorithm="fricas")

[Out]

-2*sqrt(x^3 + x)/(x^2 + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x - 1\right ) \left (x + 1\right )}{\sqrt{x \left (x^{2} + 1\right )} \left (x^{2} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-1)/(x**2+1)/(x**3+x)**(1/2),x)

[Out]

Integral((x - 1)*(x + 1)/(sqrt(x*(x**2 + 1))*(x**2 + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2} - 1}{\sqrt{x^{3} + x}{\left (x^{2} + 1\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)),x, algorithm="giac")

[Out]

integrate((x^2 - 1)/(sqrt(x^3 + x)*(x^2 + 1)), x)